Moduli space

Summary

A moduli space is a geometric space whose points correspond to the isomorphism classes of objects of some kind. The objects may be curves, vector bundles or subvarieties, for example. A fine moduli space represents the moduli functor and carries a universal family. A coarse moduli space only has the right points and is universal for maps out of the functor. Moduli spaces turn classification problems into geometry, and their geometry, such as dimension and singularities, reflects the deformations of the objects. Examples are the Grassmannian, the Hilbert scheme and the moduli of algebraic curves.